Consider the following statements. "If x is an integer then 3+2=5" and "If x is not integer then 3+2=5". Constructing truth tables for the above statements show that there is no case P is true and Q is false. So both statements are true.
Also the statement where we have substitute the above statements "If (If P then Q and If not P then Q) then Q" is true. Can we say that the whole implication or Q is a tautology? I would say no because it depends on what we have defined as "integer", "3+2=5" etc. But from the above statements we can concluded that Q is true regardless of P i.e. always true. Does it make it a tautology? Can someone help me how to distinguish them?
From what I understand (might be wrong) tautologies are about truth tables irrespective of the meaning of the statements whereas theorems are based on the meaning of the statements. Another example is the statement "If x is positive then x squared is also positivie". It is true because we can eliminate from the truth table the line with (T and F) i.e. always true but not a tautology. But also when we "search" for tautologies we search for always true statements. Do they have in common the fact that both theorems (given the set of axioms in a system) and tautologies (given the set of axioms of laws of logic) are statements that are always true?